Explaining the moon illusion
Lloyd Kaufman*† and James H. Kaufman‡
*Psychology Department, Long Island University, C.W. Post Campus, 720 Northern Boulevard, Brookville, NY 11548-1309; and ‡IBM Research Division,
Almaden Research Center, 350 Harry Road, San Jose, CA 95120
Communicated by Julian Hochberg, Columbia University, New York, NY, October 25, 1999 (received for review August 23, 1999)
An old explanation of the moon illusion holds that various cues
place the horizon moon at an effectively greater distance than the
elevated moon. Although both moons have the same angular size,
the horizon moon must be perceived as larger. More recent
explanations hold that differences in accommodation or other
factors cause the elevated moon to appear smaller. As a result of
this illusory difference in size, the elevated moon appears to be
more distant than the horizon moon. These two explanations, both
based on the geometry of stereopsis, lead to two diametrically
opposed hypotheses. That is, a depth interval at a long distance is
associated with a smaller binocular disparity, whereas an equal
depth interval at a smaller distance is associated with a larger
disparity. We conducted experiments involving artificial moons
and confirmed the hypothesis that the horizon moon is at a greater
perceptual distance. Moreover, when a moon of constant angular
size was moved closer it was also perceived as growing smaller,
which is consistent with the older explanation. Although Emmert’s
law does not predict the size– distance relationship over long
distances, we conclude that the horizon moon is perceived as larger
because the perceptual system treats it as though it is much farther
away. Finally, we observe that recent explanations substitute
perceived size for angular size as a cue to distance. Thus, they imply
that perceptions cause perceptions.
R
egardless of its elevation in the sky, the moon’s angular size
at the eye remains the same. Yet the horizon moon may
appear to be nearly twice the diameter of the elevated moon (1,
2). Over the centuries many different explanations were offered
for this immensely powerful real-world illusion (3). Today most
attention is paid to two diametrically opposed general explanations. Each explanation is exemplified by several different
versions, but all reflect two basically different views of how the
perceptual system computes size and distance. The older explanation is typified by the so-called apparent distance theory (3, 4).
Basically, it holds that the perceived distance to the moon at the
horizon is greater than that to the zenith moon. Fig. 1 shows how
the horizon moon’s greater perceived distance could cause it to
be perceived as larger. This figure illustrates a version of the
theory in which the perceived size of the moon is proportional
to its perceived distance, a relationship known as Emmert’s law.
Another version of the apparent distance theory presumes an
inherent anisotropy of visual space in which vertical distances are
underestimated as compared with distances in the horizontal
direction (5, 6). This apparent distance theory emphasizes the
angle of regard, giving much less weight to the distance cues
provided by the terrain. Although Kaufman and Rock (1) found
only a weak effect of angle of regard, they did discover that
terrain in one direction and its absence in the other plays a vital
role. All versions of the theory are based on the notion that the
horizon moon is effectively at a greater perceptual distance than
the elevated moon. Apparent distance theories presume that
perceiving the size of the moon involves the same mechanisms
as those involved in perceiving the sizes of ordinary objects.
Accurate perception of an object’s size regardless of its distance
is referred to as size constancy. Because the angular size of an
object is inversely proportional to its distance (Euclid’s law), size
constancy is possible only if distance is taken into account (7, 8).
Information regarding distance is derived from various cues such
as the convergence of the eyes and perspective (9). As such cues
500 –505 u PNAS u January 4, 2000 u vol. 97 u no. 1
Fig. 1. Regardless of its elevation, the distance between an observer (at the
center of the horizontal line) and the moon remains constant (unfilled circles).
However, a moon perceived as growing closer as its elevation increases (filled
circles), must appear as growing smaller.
are made less salient, the size of a distant object is matched more
closely to the angular size of a nearby object than its linear
size (7).
Observers asked to judge relative distances to the horizon and
elevated moons usually respond by asserting that the horizon
moon is closer. This is directly at odds with apparent distance
theories, thus creating a serious problem for them. This sizedistance paradox is not limited to the moon illusion. When any
object of constant angular size is viewed in darkness with
exaggerated convergence, it is described as both smaller and
farther (10). Such observations lead to the idea that perceived
size may not be based on perceived distance.
Recent theories postulate that the elevated moon’s small
apparent size is not related to its perceived distance. Rather, the
perceived size of the moon is the dominant cue to its distance.
This is just the opposite of the apparent distance theory.
Accommodation of the eyes is one factor purported to directly
affect the perceived size of the moon. That is, the accommodation of the eyes increases while viewing the elevated moon. It is
proposed that this leads to a reduction in the perceived size of
the moon (11).
Although accommodation is traditionally considered a cue to
distance, in this theory it has a direct effect on perceived size.
Viewing the elevated moon in empty space presumably causes
the eye to tend toward its resting focus (12). The reduced size
then acts as a cue to distance, and the moon appears to be
farther. Reduction in size caused by increased accommodation
is called accommodative micropsia. A similar reduction in size
occurs with an increase in convergence, hence the appellation
convergence micropsia. This paper raises fundamental questions
about the overall applicability of these micropsia-based theories,
which are widely cited in today’s debate about the moon illusion.
To appreciate the differences among various theories we must
consider the different meanings of size. Briefly, if a subject
matches the length in meters of a distant rod to that of a nearby
rod, as in a size constancy experiment, we refer to perceived
linear size. Alternatively, if a subject adjusts the length of a
nearby rod so that it subtends the same angle at the eye as a
†To
whom reprint requests should be addressed. E-mail: lkaufma@optonline.net.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.
Kaufman and Kaufman
tially smaller than the disparity of a point similarly perceived as
being half the distance to the elevated moon seen against an
empty sky.
(ii) The binocular disparity of a point perceived as being
halfway along the line of sight to the elevated moon is substantially smaller than the disparity of a point similarly perceived as
being half the distance to the moon seen above the terrain at the
horizon. This is precisely the opposite of Hypothesis i.
We now provide the rationale for these hypotheses.
Binocular disparity alone can signify neither distance to
objects nor the amount of depth between them. Disparity must
be calibrated by information regarding distance. This follows
from the geometry. In general, the relative disparity
| 5 @ a(E1 2 E2)] 4 [E1 3 E2].
Where a 5 distance between the eyes,
E1 5 distance in meters to one object,
E2 5 distance in meters to a more distant object.
If we let (E1 2 E2) 5 d, then the depth
d 5 @|~E1 3 E2)]ya.
In words, the depth associated with a particular binocular
disparity | is proportional to the product of the distances to the
two objects. Where the depth d is small relative to the viewing
distance, the magnitude of the depth increases approximately as
the square of the distance. This suggests that the perceived depth
associated with a given disparity is powerfully affected by
viewing distance. For example, Wallach and Zuckerman’s (21)
data show that the perceived depth associated with a relatively
small fixed disparity increases approximately with the square of
distance up to about 2 m. By contrast, the linear size of an object
of fixed angular size geometrically increases only as the first
power of its distance. The effect of distance on depth represented by disparity diminishes as distance cues are made less
effective. Unfortunately, most systematic studies of the effect of
distance on disparity have been limited to distances of about 2 m.
However, Cormack (22) demonstrated that depth perceived in
the afterimage of a stereogram increases in accord with the
geometry of stereopsis up to a distance as great as 20 m. He
further showed that depth continues to increase at greater
distances, but at a slower rate. For a constant binocular disparity
over long distances, therefore, perceived depth is a monotonic
function of perceived distance.
In addition to pictorial cues, oculomotor cues (e.g., perspective) serve to calibrate disparity (21). The weights given to these
cues by the perceptual system may differ with distance. Thus,
oculomotor cues become less effective at long distances, where
the perceptual system is more dependent on pictorial cues.
Because the threshold for detecting depth due to disparity is as
small as 1 arc-sec (23), disparity results in detection of depth
beyond the effective range of convergence. In fact, the disparity
between the moon and objects at 1 km is approximately 12
arc-sec. Depth is surely visible between objects several hundred
meters away from a subject and the moon. Beyond the range of
convergence, pictorial cues to distance must play a dominant role
in determining the perceived depth represented by relative
disparity.
Half-Distance Experiment
Method. These considerations motivated an experiment in which
subjects viewed two virtual moons adjacent to each other on the
sky. ‘‘Moons’’ were luminous disks produced on an IBM ThinkPad flat panel display mounted in a stereoscopic optical apparatus as depicted in Fig. 2. One of them (the variable moon) was
initially perceived as closer, or at the same distance as, the other
PNAS u January 4, 2000 u vol. 97 u no. 1 u 501
PSYCHOLOGY
distant rod, we refer to perceived angular size. McCready (13)
considers perceiving angular size to be equivalent to the perception of the difference in visual directions to the two ends of
the rod. It is noteworthy that an infinite number of distal rods of
different lengths and orientations may subtend the same visual
angle, and therefore visual angle per se cannot represent any
specific object.
Roscoe (11) proposes that retinal size is perceptible. Perceiving a retinal image is tantamount to perceiving the distribution
of activity across the mosaic of retinal receptors. However,
perception is of objects and not of neural activity, so strictly
speaking, image size per se may not be perceptible. Of course,
retinal size is equivalent to angular size and just as ambiguous.
Alternatively, one and only one physical object corresponds to
the perception of an object of a particular linear size, even if the
perception is invalid. Obviously, perceiving linear size is possible
if, and only if, information regarding distance to the object is
available. If this information is incorrect (e.g., when convergence
is increased by means of a prism), linear size is misperceived, but
it still corresponds to a physical object at a particular distance.
Hypothetically, if two objects were presented in the absence of
all cues to distance, perceivers may determine whether they have
the same or different retinal sizes. However, this determination
may equally well be based on the assumption that both distal
objects are at the same distances.
Despite controversy concerning the perceptibility of angular
size per se (14, 15), several theories attribute the moon illusion
to a difference in the perceived angular size of the moon (11,
16–18). As stated, some hold that a diminution of the image of
the elevated moon is a direct effect of the increase in accommodation when the moon is viewed through empty space (11,
17). Similarly, convergence purportedly increases when the eyes
roll upwards to view the elevated moon, and the terrain induces
a lessening of convergence when viewing the horizon moon (17).
Generally, theories based on oculomotor micropsia hold that the
elevated moon appears as smaller and farther than the horizon
moon.
Although Baird et al. (18) do not relate the illusion to
oculomotor micropsia, they do consider the illusion to be one of
perceived visual angle. Thus, the perceived angular size of the
moon is determined by the ratio of its angular extent to that of
its surrounding context. This causes the elevated moon to appear
as smaller and therefore as farther. Thus, perceived angular size
governs perceived distance.
Taking yet a different approach, Gregory suggests that the size
of the horizon moon is scaled by cues to distance (as in apparent
distance theories), but there are no distance cues for the elevated
moon. In that case the moon takes on a default size, which
happens to be smaller than that of the horizon moon (19). In this
context Gregory does not distinguish between apparent angular
size and apparent linear size. However, once established, the
larger size of the horizon moon operates in a ‘‘top-down’’
fashion, and modifies its perceived distance so that it seems
closer than the smaller and more distant elevated moon. Again,
perceived size acts as the predominant cue to relative perceived
distance. This is to be contrasted with Gogel’s proposal that the
elevated moon takes on a default distance, which scales the
moon’s apparent linear size (20). Because the default perceived
distance is smaller than the distance to the horizon moon, the
elevated moon is perceived as both smaller and nearer, placing
Gogel squarely in the camp of the apparent distance theorists.
The notion that perceived angular size of the moon determines
its perceived distance is diametrically opposed to the apparent
distance theory. This opposition provides an opportunity to test
two mutually exclusive hypotheses regarding the moon illusion.
These are:
(i) The binocular disparity of a point perceived as being
halfway along the line of sight to the horizon moon is substan-
Fig. 2. Schematic of apparatus. The active matrix screen of the computer was mounted horizontally in a frame 38.3 cm beneath a black screen containing two
apertures, 6.4 cm apart. Each aperture contained a 38.3-cm focal-length lens. A partially silvered front-surface 40 3 40 cm mirror was mounted at a controllable
angle above the lenses, so that subjects looking through the mirror viewed virtual images of 0.62° diameter ‘‘moons.’’ Two moons were located under each lens
to produce four virtual images at optical infinity. Subjects looking through the mirror fused the two sets of moons and saw one pair of moons in the sky.
(the reference moon). When near the horizon, the reference
moon had zero disparity, relative to objects at the horizon.
Fig. 3 is a stereogram that simulates our situation. It depicts
a landscape containing two moons, with the left-hand moon
appearing both closer and smaller than the identical right-hand
moon. The initial depth between the moons in the experiment
was produced by a small, randomly selected, binocular disparity
ranging from 0° (no depth) to 0.56° (a distance of about 6.5 m
between the subject and the variable moon). Pressing a key
increased or decreased the initial disparity in 2-arc-min steps,
thus altering only the stereoscopic depth in the display. Subjects
were asked to change the distance of the variable moon so that
it bisected the space between themselves and the distant reference moon. Another key press recorded this setting, and a new
trial was begun.
Fig. 3. Stereogram of a Tuscan landscape (photograph © 1999 by Ansen
Seale), modified, with permission, to include identical moons in each halffield. Fusing the two half-fields creates the impression that the left-hand
moon is closer and also substantially smaller than the right-hand moon, as
reported by subjects of Experiment (i) described in text. One should view the
scene by staring at a point some distance behind the page. An animation
simulating the experiment is available in the supplementary material on the
PNAS web site, www.pnas.org.
502 u www.pnas.org
One hundred trials, each beginning with a randomly selected
disparity, were conducted per subject for each condition. The
elevation of the fused images was determined by the angle of the
mirror that placed the reference moon either about 1.5° above
the horizon, or at 45° to produce an elevated moon viewed
through empty space against the sky. In both conditions, the
center of the half-image of the left-hand variable moon was
always 1.35° from the reference moon. The corresponding
distance in the right half-field was adjusted by pressing a key.
The theories discussed above imply mutually exclusive outcomes. As stated earlier, the apparent distance theory predicts
that a smaller disparity would be needed to bisect the distance
between the subject and the horizon reference moon than that
between the subject and the elevated reference moon. Theories
based on the proposition that the perceived distance to the moon
is determined by its relative size predict the opposite result,
because the reference moon above the horizon is perceived as
larger than the elevated reference moon.§
Subjects L.K. and W.H. were 70 years old and B.B. was 62
years old. W.H. had lens implants subsequent to cataract surgery.
L.K. was presbyopic, but had normal distance vision. B.B., also
presbyopic, wore contacts to correct distance vision. W.Y. was 30
years old, and F.D. was 22 years old. The experiment was
conducted on a hilltop on the C.W. Post campus of Long Island
University in Brookville, NY. Subjects looking through the
combining glass and fusing the virtual moons saw them side-byside and slightly above a very distant horizon or in an empty sky.
All of the experiments were performed in midmorning on nearly
cloudless days. The horizon was composed of hazy hills many
kilometers away, across Long Island Sound (which was itself
hidden by several kilometers of intervening terrain and treetops). When the moons were elevated there were no nearby
features or clouds. Although all of the moons were identical in
angular size (0.6°), it was evident that the horizon moon was
much larger than the elevated moon.
§An
animation simulating the moon stereogram demonstrations is available in the supplementary material on the PNAS web site, www.pnas.org.
Kaufman and Kaufman
Subject
MH
ME
SEH
SEE
W.H.
B.B.
F.D.
W.Y.
L.K.
6.293
16.877
5.409
10.071
3.132
(A) Half-distance data
21.976
0.259
26.649
1.564
23.158
0.334
48.817
0.534
20.292
0.203
1.189
1.242
1.027
1.948
0.242
W.H.
B.B.
F.D.
M.S.
L.K.
52.382
69.599
51.983
33.931
23.777
(B) Half-size data
215.308
1.385
73.209
1.611
92.365
1.230
39.964
2.953
34.618
0.482
8.703
2.012
4.463
1.501
0.797
(A) Mean disparities of elevated and horizon variables and moons relative
to reference moons in arc-min, at half-distance for each of five subjects, and
SE. (B) Mean disparities and distances of moon when adjudged to be one-half
its original size. MH, horizon moon; ME, elevated moon; SEH and SEE, estimates
of standard errors in perception of horizon and elevated moons, respectively.
Results. The mean relative disparities at which the variable moon
bisected the distance between the subject and the reference
moon are shown for each subject in Table 1A. Corresponding
estimates of the standard errors (SE) are included. The basic
data show that, on average, the angular disparity of the variable
moon relative to the reference moon is about 3.4 times greater
when the moons are elevated than when the moons are viewed
above the ground plane. This larger relative disparity corresponds to a much greater depth between the elevated reference
and the variable moons, as compared with the perceived depth
between the horizon moons.
To estimate the perceived distances to the variable moons
when subjects had bisected the space between themselves and
the reference moon, we divided the average interocular distance
of 0.064 m by the mean disparities in radians, and computed
averages across subjects. On average, subjects placed the variable
horizon moon at the same distance as an object 36.17 m away
from them. In the case of the variable elevated moon, the
corresponding distance was 8.62 m. Thus, the horizon moon was
placed approximately 4.2 times farther away than the elevated
variable moon. Despite a wide range of mean half-distance
settings, all five subjects placed the horizon variable farther away
than they did the elevated variable, with ratios ranging from 1.58
to 6.48. These results confirm Hypothesis i, which is based on
classic apparent-distance theory, and are inconsistent with Hypothesis ii, which is associated with theories holding that the
perceived size of the moon determines its perceived distance.
Discussion. How do we reconcile these results with the fact that
subjects tend to describe the larger-appearing horizon moon as
closer and not as farther? Rock and Kaufman (2) suggested that
distance cues act to scale size. Once scaled, observers use the
perceived size of the moon to make a logical choice when asked
about distances. They supported this choice by presenting large
and small moons. Nine of 10 subjects described the larger moon
as closer, regardless of whether it was elevated or at the horizon.
However, we now know that this result is also consistent with the
opposing theories. But if the moon’s perceived size determines
its perceived (as opposed to judged) distance, then the halfdistance measures should be opposite to those we obtained.
Gregory (24) noted that converging lines drawn on paper
could be seen as flat or as representing parallel lines receding
Kaufman and Kaufman
into the distance. Although subjects perceive the drawings as
flat, objects are distorted in shape or size because of the presence
of such lines. The lines act as distance cues when they promote
size constancy. Gregory suggested that these cues, presented on
flat paper, are sufficient to trigger constancy scaling, in much
same way as do the distance cues in normal scenes. However,
when confronted with conflicting information regarding the
flatness of the drawing, subjects need not be aware that different
distances result in the distortions of shape and size. This notion,
that distance cues can promote constancy without viewers being
aware that they are doing so, is similar to Rock and Kaufman’s
proposal (2).
We know that distance information scales disparity among
objects along the line of sight. Based on our data, it is evident
that differences in apparent size do not scale differences in depth
between the variable and reference moons. Hence, illusory
differences in size do not act as distance cues. They scale neither
distance nor depth. It is possible that subjects are simultaneously
processing seemingly conflicting information, i.e., illusory differences in size and actual cues to distance, in different ways.
The Moon’s Size and Its Perceived Distance
Method. Our own observations revealed that increasing the
absolute disparity of a moon results in two concomitant changes.
First, the moon appears to grow smaller. Second, it also appears
to come closer. This is easily understood in the case of the
horizon moon, because as the disparity increased, subjects were
able to see the moon at the same distances as objects that were
closer to them. However, without the benefit of objects in the
intervening space, the elevated moon exhibited the same behavior. As it grew smaller, it seemed to come closer. These observations were made in connection with our second experiment
(see Table 1B), in which we sought to test the applicability of
Emmert’s law to the perceived size of the moon.
Historically, the apparent-distance theory is related to Emmert’s law, which holds that the perceived size of an afterimage
(or any object of constant angular size) is proportional to its
perceived distance (25). However, a simple proportional relationship is unlikely at very large distances (26). In this experiment, we test for departures from Emmert’s law, by testing the
hypothesis that a moon at the half-distance should be perceived
as one-half the size of a reference moon of the same angular size.
One moon was presented to each eye. These were fused to
form a single moon that initially was at the distance of the
reference moon (zero disparity). Five subjects decreased the
distance to the moon (by increasing the absolute disparity), until
it appeared to be one-half its original size. This was done for the
moon seen just over the horizon and for the elevated moon
viewed in an empty sky.
Results. All subjects described the moon as drawing closer as its
size diminished (see results shown in Table 1B). Note that M.S.,
a 22-year-old male with normal vision, replaced W.Y., who had
served as a subject in the first experiment (Table 1 A). Table 1B
contains the mean absolute disparities at which each subject
judged the moon to be one-half its original size. This disparity
was far greater than the disparity of the variable moon at the
half-distance of the first experiment. Overall, the average distance to the horizon moon, when it was judged to be one-half its
original size, was 5.47 m. The corresponding distance of the
elevated moon was 3.65 m. Therefore, under the conditions of
this experiment, Emmert’s law does not determine the magnitude of the moon illusion.
Discussion. These results are consistent with those reported by
Enright (27), who found only an 8% reduction in the size of a
moon that was moved from a distance of about 3 km to 60 m. Our
results demonstrate that this effect becomes progressively larger,
PNAS u January 4, 2000 u vol. 97 u no. 1 u 503
PSYCHOLOGY
Table 1. Mean disparities in size of moon when perceived at half
distance, relative to reference moon (A) and when perceived as
half-size relative to original (B)
so that the moon’s size is reduced by half at distances within
about 10 m. Consequently, the size of the moon varied as some
positive function of distance. By contrast, if the moon’s size is the
predominant cue to its distance, and if then it had been seen as
growing smaller, it should have appeared to move away, or if it
had been perceived as moving closer, its perceived size should
have increased.
Although this experiment showed that perceived size is not
proportional to perceived distance, the horizon moon was still
1.5 times farther away than the elevated moon when it was judged
to be one-half its original size. If the elevated moon had initially
been perceived as more distant than the horizon moon, then it
is plausible to consider that it should have reached its half-size
when it was farther away than the horizon moon. The opposite
is the case, which is consistent with the proposition that the
perceptual system responds to the horizon moon as though it
were more distant than the elevated moon.
The perceived half-distance to the elevated variable moon was
approximately 8.6 m, more than twice the distance at which the
same moon is judged to be half its original size. Similarly, the
half-distance in the horizontal direction is about 36 m. This is
about 6 times the distance at which the horizon moon is judged
to be one-half its original size. In the second experiment, the size
reduction actually became visible almost as soon as the moon was
moved. However, subjects had to move the moon close enough
for convergence to begin to function, and this process accelerated its reduction in perceived size. Emmert’s law may hold
where oculomotor cues are given more weight (i.e., at close
distances), but the cues that operate at longer distances fail to
support it fully. Even so, the presence of terrain well beyond the
effective distance of convergence led to a more rapid reduction
in the perceived size of the horizon moon, as compared with the
elevated moon.
The Natural Moon
Finally, we briefly report observations involving the natural
moon. The authors, one naive subject, and two of the original
subjects viewed the moon at different locations, but when its
angle of elevation was about 30°. The moon was adjacent to the
top of a clearly visible nearby object at a horizontal distance of
about 15 m. The observers closed one eye after viewing the moon
binocularly. Four estimated that the moon became 20–25%
smaller, and one estimated a reduction of 10%. Two observers
viewed the moon through apertures to occlude all other objects;
monocular viewing failed to produce the reduction in moon size.
Although the natural moon was elevated, its binocular disparity
relative to a nearby object caused it to be seen as farther away
than that object. Monocular viewing eliminated this disparity,
and lacking other depth cues, observers responded as though the
moon were equidistant with the relatively close foreground
objects (20, 28).
Conclusions
We found that the perceptual system responds as though the
horizon moon were at a greater distance than the elevated moon.
This is consistent with theories that attribute the illusion to the
effects of cues signifying that the horizon moon is much more
distant than the elevated moon. Cues to distance are physical
properties of stimuli, or physiological states (e.g., convergence)
affected by those properties. Different cues lead to different
perceptual states. Thus, the term apparent in so-called apparentdistance theories is inappropriate. Rather, we suggest that the
physical cues to distance affect both perceived distance and
perceived size. The opposing apparent-visual-size theories substitute perceived size for angular size as a cue to distance. Thus,
they imply that perceptions cause perceptions.
In reporting on their impressions of size and distance, subjects
attend differently to different aspects of their perceptions. As in
504 u www.pnas.org
the case of geometrical illusions printed on paper, sometimes
these reports are at odds with some aspects of their perceptions,
although consistent with other aspects. This accounts for the
size–distance paradox. Our results contradict predictions that
are based on the assumption that the apparent angular size of the
moon determines its relative perceived distance. Mechanisms
proposed in the literature to account for differences in perceived
angular size are entirely speculative and largely unsupported by
direct evidence (see Appendix). At our latitude in New York, the
natural moon is never at a higher elevation than about 60°, and
it is most often much lower. On most occasions of viewing the
elevated moon, the terrain may therefore be barely visible in
peripheral vision. Thus, some information regarding distance to
the sky may be gleaned from peripheral vision, and is certainly
picked up through ordinary eye and head movements. Furthermore, in natural circumstances, the elevated moon may be only
a few degrees above trees, tall buildings, and hills. In addition,
the elevated moon could certainly be placed at, or somewhat
beyond, the effective distance of convergence. This distance is
approximately the same as the half-distance of our first experiment, where the elevated moon was viewed in an empty sky.
Strictly speaking, it is not true that there are no cues to the
distance of the elevated moon, although these cues are less
salient than those associated with the terrain. Consequently, we
propose that in normal circumstances, the perceived distance to
the elevated moon is a compromise between the distance of
convergence and other information picked up peripherally, or
remembered shortly after scanning the scene.
The cues most relevant to the moon illusion are those related
to the distance between the observer and other external objects.
We suggest this to be the reason that the illusion was found to
seem much weaker in some simulations. For example, when
viewed above the horizon of a projected stereogram of a terrain,
the moon is only slightly larger than when it is viewed on the
same, but dark nearby screen (2). Similarly, although depth cues
in ordinary pictures signify that the moon is more distant than
objects in the foreground, the illusion is quite weak (29).
However, these depth cues are scaled by the distance between
the observer and the page. In the natural world, the observer is
part of the scene, and the distances to points on the terrain scale
depth cues differently.
Although we attribute the illusion to a greater effective
distance to the horizon moon, we found that Emmert’s law is not
an accurate predictor of the size–distance relationship. Additional work is required to determine how different visual cues
influence the exact functional relationship between size and
distance in different perceptual regimes.
Finally, the results of these experiments are inconsistent with
theories that attribute the illusion to micropsia; they are also
inconsistent with theories that postulate a default size for the
elevated moon, which then determines its perceived distance. As
pointed out in the Appendix, there is a paucity of real evidence
to support these theories, and there is an important logical flaw.
Appendix
It is indisputable that viewing an object while accommodating
for a closer distance reduces its perceived size. But there is no
evidence that this classic effect produces a sufficiently large
difference in size to account for the moon illusion. Also, the eye
does tend toward a resting focus of about 2 m in a totally empty
visual field (12).
Roscoe, however, describes an increase in accommodation
when subjects are viewing a luminous and apparently sharp
elevated moon. Nevertheless, Roscoe considers this to be an
example of the same empty field myopia (11). Further, he reports
no data indicating that the magnitude of the moon illusion
actually varies with the magnitude of the increase in accommodation. His effect could be related to the fact that one eye was
Kaufman and Kaufman
occluded (effectively in an empty visual field), and may not have
any bearing on the moon illusion.
Although Enright also suggests a role for accommodation, his
own studies showed that the increase in accommodation vanishes
during repeated measurements (17). Holway and Boring (30)
concluded that elevation of the eyes in the head accounts for
most of the variance in the moon illusion as they measured it, but
they could not present a plausible explanation. Kaufman and
Rock (1, 4) disputed this result. Enright (17) accepted the
conclusion that elevated eyes per se will not produce the illusion,
but suggested instead that a transitory change in vergence
occurring while the eyes roll upwards results in convergence
micropsia for the elevated moon. However, there is no reported
evidence to support this conjecture.
McCready suggested that oculomotor micropsia occurs because the angle subtended by an object at a hypothetical
egocenter, which lies between the eyes and about 10 cm behind
them, is smaller than the angle subtended by the same object at
the eye (13, 16). However, this is true only for nearby objects, and
fails when the object of regard is at optical infinity. It must fail
for the same reason that the moon follows you when you move
your head laterally and does not change in size when your head
moves fore and aft.
A square increasing in linear and in angular size on a video
screen appears to move toward the subject (31). Subjects could
null this apparent motion in depth by adjusting disparity. Thus,
changing physical angular size is a cue to motion in depth.
Wallach and Frey (32) caused a luminous diamond to enlarge as
it moved along a 55-cm long track toward the observer, thus
simulating passage along a path of 367 cm. Subjects altered
convergence and accommodation in following this moving dia-
mond. Subsequent to watching this changing size, the perceived
distance of a stationary three-dimensional object was altered,
which signified that static accommodation and convergence
were recalibrated by changing size. Significantly, the index of this
altered perceived distance was a large increase in the amount of
depth related to the disparity between the nearer and farther
parts of the stationary object. Our first experiment also relied on
perceived depth represented by disparity as an index of relative
perceived distance.
In these experiments, changes in size affected perceived
distance, but the changes were in actual angular size. Theories
that attribute the moon illusion to differences in perceived size
(angular or otherwise) assume that perceived size per se is a
major cue to distance. Actual angular size differences do play an
important role in depth perception. For example, perspective
can be described as a gradient of angular sizes of similar elements
along the ground plane. Because gradients in angular size give
rise to perceptions of distance, perspective is described as a
Euclidean cue (9). By contrast, treating illusory differences in
size as cues to perceived distance implies that perceptions cause
perceptions. This is a perilous concept as explanations based on
it are essentially tautological.
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Kaufman and Kaufman
PNAS u January 4, 2000 u vol. 97 u no. 1 u 505
PSYCHOLOGY
We thank Peter Freyer and John Risko for fabricating several partially
silvered mirrors. We also thank Kevin Roche and Mark Schnee for work
on software, and Jorge Goita for fabricating parts for the heads-up
display. Walter Heimer discovered the experimental site. Ethel Matin,
Julian Hochberg, Aries Arditi, and Richard Gregory provided many
valuable comments. Wei Yang helped with the experiments. Most of all,
we express our gratitude to the late Irvin Rock, who urged us to pursue
these studies.